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Nilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

The ability to understand the surrounding environment and being able to communicate with interacting humans are important functionalities for many automated systems where visual input (e.g., images, video) and natural language input (speech or text) have to be related to each other. Possible applications are automatic image caption generation, interactive surveillance systems, or human robot interaction. In this paper, we propose algorithms for automatic responses to natural language queries about an image. Our approach uses a predefined neural net for detection of bounding boxes and objects in images, spatial relations between bounding boxes are modeled with a neural net, the queries are analyzed with a syntactic parser, and algorithms to map natural language to properties in the images are introduced. The algorithms make use of semantic similarity and antonyms. We evaluate the performance of our approach with test users assessing the quality of our system’s generated answers.

We introduce a new measure of descriptional complexity on finite automata, called the number of active states. Roughly speaking, the number of active states of an automaton A on input w counts the number of different states visited during the most economic computation of the automaton A for the word w. This concept generalizes to finite automata and regular languages in a straightforward way. We show that the number of active states of both finite automata and regular languages is computable, even with respect to nondeterministic finite automata. We further compare the number of active states to related measures for regular languages. In particular, we show incomparability to the radius of regular languages and that the difference between the number of active states and the total number of states needed in finite automata for a regular language can be of exponential order.

A synchronization problem in cellular automata has been known as the Firing Squad Synchronization Problem (FSSP), where the FSSP gives a finite-state protocol for synchronizing a large scale of cellular automata. A quest for smaller state FSSP solutions has been an interesting problem for a long time. It has been shown by Balzer (Inf Control 10:22–42, 1967), Sanders (in: Jesshope, Jossifov, Wilhelmi (eds) Proceedings of the VI international workshop on parallel processing by cellular automata and arrays, Akademie, Berlin, 1994) and Berthiaume et al. (Theoret Comput Sci 320:213–228, 2004) that there exists no 4-state FSSP solution in arrays and rings. The number four is the state lower bound in the class of FSSP protocols. Umeo et al. (Parallel Process Lett 19(2):299–313, 2009), by introducing a concept of full versus partial FSSP solutions, provided a list of the smallest 4-state symmetric powers-of-2 FSSP protocols that can synchronize any one-dimensional (1D) ring cellular automata of length (n=2^{k}) for any positive integer (k ge 1). Afterwards, Ng (in: Partial solutions for the firing squad synchronization problem on rings, ProQuest Publications, Ann Arbor, MI, 2011) also added a list of asymmetric FSSP partial solutions, thus completing the 4-state powers-of-2 FSSP partial solutions. A question whether there are any 4-state partial solutions for ring lengths other than powers-of-2 has remained open. In this paper, we answer the question by providing a new class of the smallest symmetric and asymmetric 4-state FSSP protocols that can synchronize any 1D ring of length (n=2^{k}-1) for any positive integer (k ge 2). We show that the class includes a rich variety of FSSP protocols that consists of 39 symmetric and 132 asymmetric solutions, ranging from minimum to linear synchronization time. In addition, we make an investigation into several interesting properties of those partial solutions, such as swapping general states, transposed protocols, a duality property between them, and an inclusive property of powers-of-2 solutions.

We consider the notion of a constant length queue automaton—i.e., a traditional queue automaton with a built-in constant limit on the length of its queue—as a formalism for representing regular languages. We show that the descriptional power of constant length queue automata greatly outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata. Finally, we investigate the size cost of implementing Boolean language operations on deterministic and nondeterministic constant length queue automata.

In this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.